"""
**************
Graph Matching
**************
Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent
edges; that is, no two edges share a common vertex.
`Wikipedia: Matching `_
"""
import networkx as nx
__all__ = ["min_maximal_matching"]
def min_maximal_matching(G):
r"""Returns the minimum maximal matching of G. That is, out of all maximal
matchings of the graph G, the smallest is returned.
Parameters
----------
G : NetworkX graph
Undirected graph
Returns
-------
min_maximal_matching : set
Returns a set of edges such that no two edges share a common endpoint
and every edge not in the set shares some common endpoint in the set.
Cardinality will be 2*OPT in the worst case.
Notes
-----
The algorithm computes an approximate solution fo the minimum maximal
cardinality matching problem. The solution is no more than 2 * OPT in size.
Runtime is $O(|E|)$.
References
----------
.. [1] Vazirani, Vijay Approximation Algorithms (2001)
"""
return nx.maximal_matching(G)